Finite Group Databasebeta

← #6500585

Subgroups — #6500585

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2441C211yesmaximal, characteristic
2#34212C211yesmaximal, characteristic
3#29243C212yesmaximal, complemented
4#29241C212yesmaximal, complemented
5#27680C212yesmaximal, complemented
6#26432C214yesmaximal, complemented
7#26360C214yesmaximal, complemented
8#26016C214yesmaximal, complemented
9#25973C214yesmaximal, complemented
10#24307C212yesmaximal, complemented
11#21942C211yesmaximal, characteristic
12#21934C211yesmaximal, characteristic
13#21765C211yesmaximal, characteristic
14#20911C211yesmaximal, characteristic
15#20846C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#479C2214yescomplemented
20C2×#428C2212yes—
21C2×#419C2212yes—
22C2×#407C2211yescharacteristic
23C2×#407C2211yescharacteristic
24#2441—22no—
25#2441—22no—
26#2441C2214yes—
27#2441C2214yes—
28C2×#399—21no—
29C2×#396—21no—
30#2368—22no—
31#2360C2212yes—
32#2360C2212yes—
33#2345—22no—
34#2345C2212yes—
35Q8×C22⋊C4C2212yes—
36#2329—21no—
37#2329—21no—
38#2315—21no—
39C4×C22⋊Q8C2212yes—
40C4×C22⋊Q8C2212yes—
41C2×#389—21no—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#387—21no—
45C2×#384C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48#2273—21no—
49C2×C4×C22⋊C4C2211yescharacteristic
50#2099C2214yescomplemented
51#2096C2214yescomplemented
52#2088C2214yescomplemented
53#2087C2214yescomplemented
54#2068C2214yescomplemented
55#2068C2214yescomplemented
56#2059C2214yescomplemented
57#2026C2214yescomplemented
58#1989C2214yescomplemented
59#1948C2214yescomplemented
60#1921C2214yescomplemented
61#1918C2214yescomplemented
62#1894C2214yes—
63#1873C2214yes—
64#1872C2212yes—
65#1872C2212yes—
66#1872C2212yes—
67#1872C2212yes—
68#1865C2214yescomplemented
69#1863C2214yescomplemented
70#1854C2214yes—
71#1853C2212yes—
72#1853C2212yes—
73#1853C2212yes—
74#1852C2212yes—
75#1852C2212yes—
76#1852C2212yes—
77#1851C2214yes—
78#1851C2214yes—
79#1823C2212yes—
80#1809C2214yes—
81#1803C2214yescomplemented
82#1797C2212yes—
83#1797C2212yes—
84#1787C2212yes—
85#1736C2212yes—
86#1736C2212yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×M16C2312yes—
90C22×M16C2312yes—
91C2×#125—22no—
92C2×#125—44no—
93C2×#125C2314yes—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8Q818yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2312yes—
102C2×#117—21no—
103C2×#117—21no—
104C2×#117—44no—
105C2×#117C2312yes—
106C2×#117C2312yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4D411yescharacteristic
126C22×C22⋊C4D411yescharacteristic
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#479—48no—
131#428—44no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2314yes—
138#419—44no—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143#407—24no—
144#407—24no—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149#399—24no—
150#399—24no—
151#396—24no—
152#396—24no—
153#391—22no—
154#391—22no—
155#391C2312yes—
156#391C2312yes—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—24no—
162#389—24no—
163#389C2312yes—
164#389C2314yes—
165#389C2314yes—
166#389C2312yes—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—24no—
174#387—24no—
175#385—22no—
176#385—22no—
177#385—22no—
178#385—22no—
179#384—22no—
180#384C2314yes—
181#380—21no—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—22no—
186#380—22no—
187#380C2312yes—
188#380C2312yes—
189#380C2312yes—
190#380C2312yes—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4C2312yes—
204C4×C22⋊C4C2312yes—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95D411yescharacteristic
210C2×#95D411yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#336C2314yes—
220#336C2314yes—
221C25C2×D411yescharacteristic
222C22×Q8—21no—
223C22×Q8—22no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8C2×Q814yes—
227C22×Q8C2412yes—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×Q814yes—
246C23×C4#4114yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×M16—24no—
251C2×M16—24no—
252C2×M16—42no—
253C2×M16—42no—
254C2×M16—42no—
255C2×M16—42no—
256C22×C8C2414yes—
257C22×C8C2414yes—
258#125—28no—
259#125—44no—
260#125—48no—
261C22⋊Q8—28no—
262C22⋊Q8—28no—
263C22⋊Q8—24no—
264C22⋊Q8—24no—
265C22⋊Q8C2×Q818yes—
266C22⋊Q8C2×Q818yes—
267C4×Q8—44no—
268C4×Q8—44no—
269#117—24no—
270#117—24no—
271#117—42no—
272#117—42no—
273#117—44no—
274#117—44no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4C2×Q814yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—42no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—82no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—44no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2412yes—
346C2×C42C2412yes—
347#107—28no—
348#107—44no—
349#106—44no—
350#106—44no—
351C4⋊C8—44no—
352C4⋊C8—44no—
353#103—28no—
354#103—44no—
355#103—44no—
356#103—44no—
357C22⋊C8—24no—
358C22⋊C8—24no—
359C22⋊C8—24no—
360C22⋊C8—24no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—22no—
380C24—22no—
381C24—41no—
382C24—41no—
383C24—41no—
384C24—41no—
385C24C22⋊Q814yes—
386C24C22×D411yescharacteristic
387C24C22×D411yescharacteristic
388C24C22×D411yescharacteristic
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—48no—
394C2×Q8—81no—
395C2×Q8—81no—
396C2×Q8—84no—
397C2×Q8C22×Q814yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4C22⋊Q814yes—
456C22×C4C22⋊Q814yes—
457C22×C4C22⋊Q814yes—
458C22×C4#13714yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×Q812yes—
467C22×C4C22×Q812yes—
468C22×C4C22×Q812yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2511yescharacteristic
473M16—48no—
474M16—84no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C4⋊C4—24no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—42no—
496C4⋊C4—48no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—81no—
503C4⋊C4—81no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4C22×Q814yes—
509C4⋊C4H(5,𝔽2)14yes—
510C22⋊C4—28no—
511C22⋊C4—28no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C42—24no—
523C42—24no—
524C42—42no—
525C42—42no—
526C42—44no—
527C42—44no—
528C42—82no—
529C42—82no—
530C42—82no—
531C42—82no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—28no—
549C23—44no—
550C23—44no—
551C23—82no—
552C23—82no—
553C23—82no—
554C23—82no—
555C23C2×C22⋊Q812yes—
556C23C2×C22⋊Q812yes—
557C23C2×C22⋊Q812yes—
558C23C2×#13712yes—
559C23#57512yes—
560C23C23×D411yescharacteristic
561Q8—28no—
562Q8—44no—
563Q8—84no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—162no—
625C2×C4—162no—
626C2×C4C2×C22⋊Q812yes—
627C2×C4C2×C22⋊Q812yes—
628C2×C4C2×C22⋊Q812yes—
629C2×C4C2×#13714yes—
630C2×C4C23×Q811yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×H(5,𝔽2)12yes—
633C2×C4#58512yes—
634C8—44no—
635C8—44no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—88no—
657C22#307814yes—
658C22C22×C22⋊Q811yescharacteristic
659C22#348411yescharacteristic
660C22#348511yescharacteristic
661C22C22×#13711yescharacteristic
662C22C2×#57511yescharacteristic
663C22#358511yescharacteristic
664C22#358611yescharacteristic
665C4—24no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—44no—
670C4—42no—
671C4—44no—
672C4—42no—
673C4—44no—
674C4—42no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4#350112yes—
681C4#350512yes—
682C2—24no—
683C2—42no—
684C2—42no—
685C2#3534911yescharacteristic
686C2C2×#307811yescharacteristic
687C2#6049311yescharacteristic
688C2#6157511yescharacteristic
689C2#6158811yescharacteristic
690C2#6170511yescharacteristic
691C2#6171811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #29243C2semidirect
4. #29241C2semidirect
5. #27680C2semidirect
6. #26432C2semidirect
7. #26360C2semidirect
8. #26016C2semidirect
9. #25973C2semidirect
10. #24307C2semidirect
19. C2×#479C22semidirect
50. #2099C22semidirect
51. #2096C22semidirect
52. #2088C22semidirect
53. #2087C22semidirect
54. #2068C22semidirect
55. #2068C22semidirect
56. #2059C22semidirect
57. #2026C22semidirect
58. #1989C22semidirect
59. #1948C22semidirect
60. #1921C22semidirect
61. #1918C22semidirect
68. #1865C22semidirect
69. #1863C22semidirect
81. #1803C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#24412
#342122
#292432
#292412
#276802
#264322
#263602
#260162
#259732
#243072
#219422
#219342
#217652
#209112
#208462